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In the simply typed lambda calculus extended with recursive types, fixed-point operators can be written, but the type of a "useful" fixed-point operator (one whose application always returns) may be restricted.
System T extends the simply typed lambda calculus with a type of natural numbers and higher order primitive recursion; in this system all functions provably recursive in Peano arithmetic are definable.
That means that there are computable functions that cannot be defined in the simply typed lambda calculus (and similarly there are computable functions that cannot be computed in the calculus of constructions or System F).
这意味着有可计算函数不能在简单类型的 lambda 演算中定义(类似的有可计算函数不能在构造演算或系统 F 中计算)。
For instance, the simply typed lambda calculus corresponds to the internal language of cartesian closed categories.
例如,简单类型λ演算 对应于笛卡儿闭范畴 (CCC)的内部语言。
The term simple type is also used to refer to extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF).
Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry-Howard isomorphism and they can be considered as the internal language of classes of categories, e.g. the simply typed lambda calculus is the language of Cartesian closed categories (CCCs).
Based on work by Berardi on pure type systems, Henk Barendregt proposed the Lambda cube to systematize the relations of pure typed lambda calculi (including simply typed lambda calculus, System F, LF and the calculus of constructions).
Simply typed lambda calculus which is a higher-order logic
简单类型λ演算 ,一种高阶逻辑 ;
Simply typed lambda calculus which is a higher-order logic; intuitionistic type theory; system F; LF is often used to define other type theories; calculus of constructions and its derivatives.
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